#457585

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,21)(17,22)(18,34)(19,24)(20,28)(23,36)(26,43)(27,39)(29,41)(30,42)(31,44)(32,40)(33,37)(35,38);(1,2)(3,4)(13,15)(14,20)(16,30)(17,27)(18,35)(19,29)(21,22)(23,41)(24,38)(25,34)(26,31)(28,36)(32,33)(37,39)(40,43)(42,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,26)(17,29)(18,27)(19,32)(20,37)(22,41)(23,44)(24,40)(25,43)(28,33)(30,35)(31,36)(34,39)(38,42);(13,17,19,26)(14,22,24,43)(15,27,29,31)(16,28,32,35)(18,30,36,33)(20,21,38,40)(23,37,34,42)(25,44,41,39)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364781);